The present invention relates to balanced mixers for multiplying a plurality of signals to derive the product thereof and, more particularly, to balanced mixers for multiplying signals generated by discrete approximation.
Balanced mixers are known in the art and are widely used in signal generation and signal detection equipment. An example of the function of a balanced mixer is where two pure sine waves of frequencies .omega..sub.1 and .omega..sub.2 are multiplied by the mixer to provide a mixer product signal having components at the sum and difference, .omega..sub.1 +.omega..sub.2 and .vertline..omega..sub.2 -.omega..sub.1 .vertline., respectively, of the frequencies of the sine waves being mixed.
In signal generation equipment, such as frequency synthesizers, the signals to be mixed are frequently approximated by discrete signals to simplify the generation process. For example, a sine wave may be approximated to first order by a square wave of the same frequency. More accurate approximations of continuous, periodic signals may be achieved by using a weighted superposition of multiple discrete signals, such as square waves, having appropriate phase delays. The circuits used to provide such discrete approximations of periodic, continuous signals are known and generally include a counter circuit for providing a set of discrete (digital) signals having appropriate phase relationships between one another and a weighting and summing network for assigning an appropriate weight to each discrete signal and adding each weighted signal to derive the approximated signal. Therefore, the mixing of discrete approximations of signals involves the mixing of weighted discrete signals (i.e., weighted digital signals that have only two discrete levels).
Digital signals are most appropriately mixed by an EXCLUSIVE NOR gate, which is the digital equivalent of the balanced mixer. An EXCLUSIVE OR gate will serve equally well as a mixer for digital signals, with the only difference being an inconsequential phase reversal in the output signal provided by the gate. Owing to the simplicity of construction of the EXCLUSIVE NOR and EXCLUSIVE OR gates, when compared with that of the conventional diode bridge balanced mixer, it would be highly advantageous to use such components in a balanced mixer for signals that are approximated by weighted digital signals.
In a typical heterodyne synthesizer arrangement, the frequencies .omega..sub.1 and .omega..sub.2 of the signals being mixed are selected such that the difference between those frequencies .vertline..omega..sub.1 -.omega..sub.2 .vertline. is equal to the desired synthesis frequency .omega..sub.0 and that .omega..sub.1 and .omega..sub.2 are both substantially higher than .omega..sub.0.
Ordinarily, a low-pass filter having a cutoff frequency .omega..sub.0 is required to suppress the unwanted components of the mixer product signal, at least the component at the sum frequency .omega..sub.1 +.omega..sub.2. The difficulty of implementing such a low-pass filter depends on such factors as the closeness of the frequencies .omega..sub.1 and .omega..sub.2 to the synthesized frequency .omega..sub.0, the required spectral purity of the synthesized signal and the accuracy with which the mixer function approximates true multiplication of input signals.
The mixing of two square waves S.sub.1 and S.sub.2 having frequencies .omega..sub.1 and .omega..sub.2, respectively, by an EXCLUSIVE OR gate to obtain a product signal S.sub.0 is graphically illustrated in FIG. 1, which shows graphs of the amplitudes .vertline.S.sub.1 .vertline., .vertline.S.sub.2 .vertline. and .vertline.S.sub.0 .vertline. of the input signals S.sub.1 and S.sub.2 and the product signal S.sub.0, respectively, versus frequency .omega.. The spectral distribution of the product signal S.sub.0 is derived by making a Fourier series expansion of the input signals S.sub.1 and S.sub.2 and multiplying the Fourier components of one input signal with those of the other. For simplicity of illustration, the graph of .vertline.S.sub.0 .vertline. versus frequency .omega. in FIG. 1 takes into account only the first three Fourier components of S.sub.1 and S.sub.2. A mixing chart 200 tabulating the frequencies and the relative amplitudes of the various products of the first three Fourier components of S.sub.1 and S.sub.2 is shown in FIG. 2. Each block of the mixing chart 200 contains the sum and difference frequencies resulting from the multiplication of the Fourier components corresponding to the row and column in which the block is situated. The number at the upper left corner of each block denotes the relative amplitude of the components within the block.
As is apparent from the graph of .vertline.S.sub.0 .vertline. versus frequency .omega. in FIG. 1, the output signal from an EXCLUSIVE OR gate mixer contains numerous and powerful unwanted spectral components. As such, burdensome requirements are placed on the low-pass filter to reject the unwanted components. In general, the most troublesome unwanted components in the mixer product S.sub.0 are those having frequencies n.vertline..omega..sub.1 -.omega..sub.2 .vertline. and corresponding relative amplitudes 1/n.sup.2. Since the desired component of the mixer product S.sub.0 has frequency .vertline..omega..sub.1 -.omega..sub.2 .vertline., these undesired components of S.sub.0 for n.noteq.1 appear as the harmonics of the desired component.
In the example illustrated in FIG. 1, the first unwanted harmonic component in the mixer product S.sub.0 is n=3 having a relative amplitude of 1/9. If a fixed-cutoff-frequency low-pass filter is used to reject all unwanted harmonics, it must have a cutoff frequency .omega..sub.C that is less than three times the lowest synthesis frequency .omega..sub.L. However, the use of such a filter also limits the highest synthesis frequency .omega..sub.H to be less than .omega..sub.C and the ratio of the highest to lowest synthesis frequency .omega..sub.H /.omega..sub.L to less than three. Consequently, the use of the EXCLUSIVE OR gate as the mixer not only imposes burdensome requirements on the low-pass filter for removing the unwanted components in the mixer product but also severely restricts the frequency tuning range of the synthesizer.
An alternative approach which imposes less stringent filtering requirements for the removal of unwanted components of the mixer product while still taking advantage of the digital properties of discretely approximated signals is to mix a two step approximation, i.e., a square wave, with a pure sine wave. The mixing of a square wave signal S.sub.1 of the frequency .omega..sub.1 and a sine wave signal S.sub.2 of frequency .omega..sub.2 to provide a mixer product signal S.sub.0 is graphically depicted in FIG. 3, which shows graphs of the amplitudes .vertline.S.sub.1 .vertline., .vertline.S.sub.2 .vertline. and .vertline.S.sub.0 .vertline. of those signals versus frequency .omega.. The mixer product signal S.sub.0 is derived by expanding the square wave signal S.sub.1 in a Fourier series and by multiplying each of the Fourier components of S.sub.1 by the sine wave signal S.sub.2. For simplicity of illustration, the graphs of .vertline.S.sub.1 .vertline. and .vertline.S.sub.0 .vertline. versus frequency take into account only the first three Fourier components of the signal S.sub.1. A mixing chart 400 tabulating the frequencies and relative amplitudes of the various products of the first three Fourier components of S.sub.1 and S.sub.2 is shown in FIG. 4. From the graph of .vertline.S.sub.0 .vertline. versus frequency in FIG. 4, it may be noted that when the square wave S.sub.1 is mixed with the sine wave S.sub.2, the separation between the nearest unwanted component of the mixer product signal S.sub.0 at frequency 3.omega..sub.1 -.omega..sub.2 and the desired component at frequency .vertline..omega..sub.1 -.omega..sub.2 .vertline. is much greater than in the case where two square waves are mixed. As such, the filtering requirements for removing the unwanted components of the mixer product are more easily met, and the desired component of the mixer product signal can have a wider frequency tuning range.
However, the mixing of a discretely approximated signal with a purely sinusoidal signal has the drawback in that a mixer for such signals is difficult to realize. The multiplication of a digital signal with a sinusoidal signal may be done with a special mixer which is schematically illustrated in FIG. 5. Referring to FIG. 5, the mixer 500 includes an analog switch 501, which is responsive to the digital signal S.sub.1 for coupling either the sinusoidal signal S.sub.2 or its inverse to the mixer output 502, depending on whether the digital signal is at a "1" or a "0" logic level, respectively. The circuit of FIG. 5 is difficult to implement in that it requires a nearly ideal analog switch 501 and inverting amplifier 502 in order to achieve a high degree of suppression of the input signals S.sub.1 and S.sub.2 and to provide a good approximation of true multiplication of those signals.
Accordingly, a need clearly exists for a balanced mixer for discretely approximated signals which can be implemented primarily with digital components and which provides a desired mixer product component that is well separated from unwanted components so as to facilitate the removal of the unwanted components by filtering and to allow the desired component be tuned over a wide frequency range.